Kevin E. M. Church

Université de Montréal | UdeM · Center for Mathematical Research

A time-delayed SIR model with general nonlinear incidence rate, pulse vaccination and temporary immunity is developed. The basic reproduction number is derived and it is shown that the disease-free periodic solution generically undergoes a transcritical bifurcation to an endemic periodic solution as the vaccination coverage drops below a critical l...

This monograph presents the most recent progress in bifurcation theory of impulsive dynamical systems with time delays and other functional dependence. It covers not only smooth local bifurcations, but also some non-smooth bifurcation phenomena that are unique to impulsive dynamical systems. The monograph is split into four distinct parts, independ...

We consider spatially homogeneous Ho\v{r}ava-Lifshitz (HL) models that perturb General Relativity (GR) by a parameter $v\in (0,1)$ such that GR occurs at $v=1/2$. We describe the dynamics for the extremal case $v=0$, which possess the usual Bianchi hierarchy: type $\mathrm{I}$ (Kasner circle of equilibria), type $\mathrm{II}$ (heteroclinics that i...

We present an implicit method of steps for differential equations with state-dependent delays and validated numerics to rigorously enclose solutions of initial-value problems. Our approach uses a combination of contraction mapping arguments based on a Newton-Kantorovich type theorem and piecewise polynomial interpolation. Completing multiple steps...

Based on the centre manifold theorem for impulsive delay differential equations, we derive impulsive evolution equations and boundary conditions associated to a concrete representation of the centre manifold in Euclidean space, as well as finite-dimensional impulsive differential equations associated to the evolution on these manifolds. Though the...

Throughout the last two centuries, vaccines have been helpful in mitigating numerous epidemic diseases. However, vaccine hesitancy has been identified as a substantial obstacle in healthcare management. We examined the epidemiological dynamics of an emerging infection under vaccination using an SVEIR model with differential morbidity. We mathematic...

We raise concerns about controllers’ robustness in simple reinforcement learning benchmark problems. We focus on neural network controllers and their low neuron and symbolic abstractions. A typical controller reaching high mean return values still generates an abundance of persistent low-return solutions, which is a highly undesirable property, eas...

We raise concerns about controllers' robustness in simple reinforcement learning benchmark problems. We focus on neural network controllers and their low neuron and symbolic abstractions. A typical controller reaching high mean return values still generates an abundance of persistent low-return solutions, which is a highly undesirable property, eas...

Link: https://2023.smb.org/MEPI/PS02-MEPI-3.html

We present a computer-assisted approach to prove the existence of Hopf bubbles and degenerate Hopf bifurcations in ordinary and delay differential equations. We apply the method to rigorously investigate these nonlocal orbit structures in the FitzHugh–Nagumo equation, the extended Lorenz-84 model and a time-delay SI model.

We consider spatially homogeneous Ho\v{r}ava-Lifshitz (HL) models that perturb General Relativity (GR) by a parameter $v\in (0,1)$ such that GR occurs at $v=1/2$. We describe the dynamics for the extremal case $v=0$, which possess the usual Bianchi hierarchy: type $\mathrm{I}$ (Kasner circle of equilibria), type $\mathrm{II}$ (heteroclinics that in...

We present a computational framework to complete computer-assisted proofs involving functions that arise from the training of deep neural networks. Our approach does not require global smoothness of nonlinearities, instead requiring functions be implemented according to a specification we call domain self-validation. This allows a posteriori valida...

We prove that under fairly natural conditions on the state space and nonlinearities, it is typical for an impulsive differential equation with state-dependent delay to exhibit non-uniqueness of solutions. On a constructive note, we show that uniqueness of solutions can be recovered using a Winston-type condition on the state-dependent delay. Irresp...

We present an implicit method of steps for differential equations with state-dependent delays and validated numerics to rigorously enclose solutions of initial-value problems. Our approach uses a combination of contraction mapping arguments based on a Newton-Kantorovich type theorem and piecewise polynomial interpolation. Completing multiple steps...

We present a computer-assisted approach to prove the existence of Hopf bubbles and degenerate Hopf bifurcations in ordinary and delay differential equations. We apply the method to rigorously investigate these nonlocal bifurcation structures in the FitzHugh-Nagumo equation, the extended Lorenz-84 model and a time-delay SI model.

We develop a rigorous numerical method for periodic solutions of impulsive delay differential equations, as well as parameterized branches of periodic solutions. We are able to compute approximate periodic solutions to high precision and with computer-assisted proof, verify that these approximate solutions are close to true solutions with explicitl...

We develop validated numerical methods for the computation of Floquet multipliers of equilibria and periodic solutions of delay differential equations, as well as impulsive delay differential equations. Using our methods, one can rigorously count the number of Floquet multipliers outside a closed disc centered at zero or the number of multipliers c...

We propose a rigorously validated numerical method to prove the existence of Hopf bifurcations in functional differential equations of mixed type. The eigenvalue transversality and steady state conditions are verified using the Newton-Kantorovich theorem. The non-resonance condition and simplicity of the critical eigenvalues are verified by either...

Using rigorous numerical methods, we prove the existence of 608 isolated periodic orbits in a gravitational billiard in a vibrating unbounded parabolic domain. We then perform pseudo-arclength continuation in the amplitude of the parabolic surface’s oscillation to compute large, global branches of periodic orbits. These branches are themselves prov...

We prove that under fairly natural conditions on the state space and nonlinearities, it is typical for an impulsive differential equation with state-dependent delay to exhibit non-uniqueness of solutions. On a constructive note, we show that uniqueness of solutions can be recovered using a Winston-type condition on the state-dependent delay. Irresp...

We propose an impulsive stabilization method for delay equations based on a reduction to the centre-unstable manifold. Our approach does not make use of the constructions of Lyapunov’s direct method, such as Lyapunov functions or functionals. Necessary and sufficient conditions for when a system can be stabilized by impulses are derived based on th...

A follow-up to Chapter 5, this chapter is devoted to computational aspects of centre manifold theory. This includes a concrete representation of the centre manifold in Euclidean space, Taylor expansions, and an explicit ordinary impulsive differential equation for the dynamics on the manifold.

The centre manifold reduction provides a framework in which bifurcations of fixed points and periodic solutions can be studied. In this section we will explain how the centre manifold reduction can be adapted to take into account parameters, prove two generic bifurcation patterns and present a general recipe for how one might study smooth local bif...

In this final chapter, we study an in-host viral infection model with impulsive drug treatment. We first restrict to the disease-free subspace and prove the existence of a disease-free periodic solution and a compact, invariant global attractor. Next, we move to the full model and show that solutions are bounded and the positive orthant is invarian...

Here we analyze a susceptible-infected-removed (SIR) model with pulse vaccination and temporary immunity modeled by a discrete delay. We complete an analysis of the disease-free equilibrium and the transcritical bifurcation of periodic orbits that occurs as the basic reproduction number crosses through unity. A cylinder bifurcation at the endemic p...

In the theory of impulsive dynamical systems, impulses are often interpreted as idealized discrete jumps associated with a process that is continuous in time but occurs on a negligibly small time scale. The intuition is that one can ignore the transient small-time intermediate dynamics and consider only the change in state. In this chapter, we expl...

Moving averages and other functional forecasting models are used to inform policy in pandemic response. In this paper, we analyze an infectious disease model in which the contact rate switches between two levels when the moving average of active cases crosses one of two thresholds. The switching mechanism naturally forces the existence of periodic...

We propose a rigorously validated numerical method to prove the existence of Hopf bifurcations in functional differential equations of mixed type. The eigenvalue transversality and steady state conditions are verified using the Newton-Kantorovich theorem. The non-resonance condition and simplicity of the critical eigenvalues are verified by either...

We study the dynamics resulting from impulsive damping or driving forces applied to a classical rigid pendulum. The impulse effect adjusts the angular velocity at fixed times based on either a point measurement or an average. In the latter, the result is that the impulse effect contains a delayed term. We complete a stability analysis and prove the...

This chapter will be devoted to the invariant manifold theory of impulsive differential equations. At the theoretical level, we will assume only that the reference bounded solution has exponential trichotomy, but when we move into computational aspects we will assume that the dynamics are periodic. This will allow us to take advantage of the Floque...

This chapter presents analogues of the classical codimension-one bifurcations of flows and maps. Specifically, this includes analogues of the fold (saddle-node), period-doubling, and Hopf (Neimark-Sacker) bifurcation. Additionally, we present the transcritical and pitchfork bifurcations.

We prove the existence of a transcritical bifurcation in the pulse-harvested Hutchinson equation. Specifically, we prove the bifurcation with respect to the period of impulse effect. The proof requires a reduction to a non-smooth centre manifold and a careful fixed-point argument.

We study bifurcations in a stage-structured predator prey system with pulsed birth. We determine the stability of the extinction equilibrium. Next, we establish the existence and uniqueness of a predator-free periodic solution and determine its stability. We then prove that the latter periodic solution can pass through the extinction equilibrium by...

We discuss the existence and smoothness of unstable, stable and centre-stable manifolds, thereby establishing the classical hierarchy of invariant manifolds for impulsive functional differential equations.

This chapter contains a proof of the principle of linearized stability for nonlinear impulsive functional differential equations, in addition to some auxiliary results on smooth dependence on initial conditions.

In this chapter, we cover existence and uniqueness of solutions, evolutions families, phase-space decompositions and the variation-of-constants formula for linear impulsive functional differential equations.

This chapter contains proofs of the existence, smoothness, and reduction principle of centre manifolds for impulsive functional differential equations. We also derive the abstract dynamics equations on the centre manifold.

In this chapter, we develop theoretical and computational aspects of Floquet theory for periodic linear systems.

In this chapter we will be interested in bifurcations that result from two “non-smooth” phenomena: perturbations in the sequence of impulses andcrossings of discrete delays across impulse times. perturbations in the sequence of impulses and crossings of discrete delays across impulse times.

This chapter is devoted to some preliminary background on ordinary impulsive differential equations. This includes existence and uniqueness of solutions and dependence on initial conditions and parameters.

This chapter contains the essential theory of linear impulsive differential equations. This includes the variation-of-constants formula, Floquet theory, and stability.

In this chapter we will discuss some methods of proving stability for nonlinear systems.

We develop a rigorous numerical method for periodic solutions of impulsive delay differential equations, as well as parameterized branches of periodic solutions. We are able to compute approximate periodic solutions to high precision and with computer-assisted proof, verify that these approximate solutions are close to true solutions with explicitl...

Moving averages and other functional forecasting models are used to inform policy in pandemic response. In this paper, we analyze an infectious disease model in which the contact rate switches between two levels when the moving average of active cases crosses one of two thresholds. The switching mechanism naturally forces the existence of periodic...

We develop validated numerical methods for the computation of Floquet multipliers of equilibria and periodic solutions of delay differential equations, as well as impulsive delay differential equations. Using our methods, one can rigorously count the number of Floquet multipliers outside a closed disc centered at zero or the number of multipliers c...

ISIM1s consists of a few MATLAB functions and a script that can be used to derive stabilizing impulsive controllers for delay differential equations. This document serves as both a manual and tutorial on the functionality of the ISIM1s package. Brief background on the theoretically guaranteed stabilization scenario are provided before the primary M...

A novel technque of robust stabilization and bifurcation suppression is proposed. The proposed method, the centre probe method (CPM), stabilizes an equilibrium point of a delay differential equation at a bifurcation point by introducing an impulsive controller that minimizes a given cost functional. The cost functional can weight certain structural...

The existence and smoothness of centre manifolds and a reduction principle are proven for impulsive delay differential equations. Several intermediate results of theoretical interest are developed, including a variation of constants formula for linear equations in the phase space of right-continuous regulated functions, linear variational equation...

The celebrated Hartman-Grobman theorem for ordinary differential equations states that the phase portrait nearby a hyperbolic equilibrium point of a nonlinear system is equivalent to that of its linearization by a conjugation. Hartman and Grobmans theorem has been extended in numerous ways to accommodate more general classes of dynamical systems. F...

In this article, we present a systematic approach to bifurcation analysis of impulsive systems with autonomous or periodic right-hand sides that may exhibit delayed impulse terms. Methods include Lyapunov–Schmidt reduction and center manifold reduction. Both methods are presented abstractly in the context of the stroboscopic map associated to a giv...

The time-scale tolerance for linear ordinary impulsive differential equations is introduced. How large the time-scale tolerance is directly reflects the degree to which the qualitative dynamics of the linear impulsive system can be affected by replacing the impulse effect with a continuous (as opposed to discontinuous, impulsive) perturbation, prod...

In this article, we examine nonautonomous bifurcation patterns in nonlinear systems of impulsive differential equations. The approach is based on Lyapunov–Schmidt reduction applied to the linearization of a particular nonlinear integral operator whose zeroes coincide with bounded solutions of the impulsive differential equation in question. This le...

There is an urgent need for more understanding of the effects of surveillance on malaria control. Indoor residual spraying has had beneficial effects on global malaria reduction, but resistance to the insecticide poses a threat to eradication. We develop a model of impulsive differential equations to account for a resistant strain of mosquitoes tha...

Analogues of the classical existence and uniqueness of solutions are proven for impulse extension equations. An exposition on matrix solutions, their properties and Floquet's theorem for periodic linear systems is provided, including applications to stability. Where applicable, comparison is made to the analoguous results from impulsive differentia...

We introduce the notion of impulse extension equations for linear fixed-time impulsive differential equations (IDEs) with strictly inhomogeneous impulses. These differential equations can be thought of as representing the underlying processes for which such linear fixed-time IDEs are a limiting case. We will establish basic existence and uniqueness...